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SUMMARY:X-ray Tomography and Discretization of Inverse Problems - Lassas\,
  M (University of Helsinki)
DTSTART:20110825T084500Z
DTEND:20110825T093000Z
UID:TALK32488@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:In this talk we consider the question how inverse problems pos
 ed for continuous objects\, for instance for continuous functions\, can be
  discretized. This means the approximation of the problem by infinite dime
 nsional inverse problems. We will consider linear inverse problems of the 
 form $m=Af+psilon$. Here\, the function $m$ is the measurement\, $A$ is a
  ill-conditioned linear operator\, $u$ is an unknown function\, and $psil
 on$ is random noise.\nThe inverse problem means determination of $u$ when 
 $m$ is given.\nIn particular\, we consider the X-ray tomography with spars
 e or limited angle measurements where $A$ corresponds to integrals of the 
 attenuation function $u(x)$ over lines in a family $Gamma$.\nThe tradition
 al solutions for the problem include the generalized Tikhonov regularizati
 on and the estimation of $u$ using Bayesian methods. To solve the problem 
 in practice $u$ and $m$ are discretized\, that is\, approximated by vector
 s in an infinite dimensional vector space. We show positive results when t
 his approximation can successfully be done and consider examples of proble
 ms that can appear. As an example\, we consider the total variation (TV) a
 nd Besov norm penalty regularization\, the Bayesian analysis based on tota
 l variation prior and Besov priors.\n\n
LOCATION:Seminar Room 1\, Newton Institute
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